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TR20-094 | 24th June 2020 08:24
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#### Is it possible to improve Yao’s XOR lemma using reductions that exploit the efficiency of their oracle?

**Abstract:**
Yao's XOR lemma states that for every function $f:\set{0,1}^k \ar \set{0,1}$, if $f$ has hardness $2/3$ for $P/poly$ (meaning that for every circuit $C$ in $P/poly$, $\Pr[C(X)=f(X)] \le 2/3$ on a uniform input $X$), then the task of computing $f(X_1) \oplus \ldots \oplus f(X_t)$ for sufficiently large $t$ has hardness $\half +\epsilon$ for $P/poly$.

Known proofs of this lemma cannot achieve $\epsilon=\frac{1}{k^{\omega(1)}}$, and even for $\epsilon=\frac{1}{k}$, we do not know how to replace

$P/poly$ by AC$^0[\textsc{parity}]$ (the class of constant depth circuits with the gates $\set{\textsc{and,or,not,parity}}$ of unbounded fan-in).

Recently, Grinberg, Shaltiel and Viola (FOCS 2018) (building on a sequence of earlier works) showed that these limitations cannot be circumvented by \emph{black-box reductions}. Namely, by reductions $\Red^{(\cdot)}$ that given oracle access to a function $D$ that violates the conclusion of Yao's XOR lemma, implement a circuit that violates the assumption of Yao's XOR lemma.

There are a few known reductions in the related literature on worst-case to average case reductions that are \emph{non-black box}. Specifically, the reductions of Gutfreund, Shaltiel and Ta Shma (Computational Complexity 2007) and Hirahara (FOCS 2018)) are ``class reductions'' that are only guaranteed to succeed when given oracle access to an oracle $D$ from some efficient class of algorithms. These works seem to circumvent some black-box impossibility results.

In this paper we extend the previous limitations of Grinberg, Shaltiel and Viola to class reductions, giving evidence that class reductions cannot yield the desired improvements in Yao's XOR lemma. To the best of our knowledge, this is the first limitation on reductions for hardness amplification that applies to class reductions.

Our technique imitates the previous lower bounds for black-box reductions, replacing the inefficient oracle used in that proof, with an efficient one that is based on limited independence, and developing tools to deal with the technical difficulties that arise following this replacement.